One area where mfc has been employed as a way to both detect and quantify compositionality is the field of emergent communication [1]}, [2]}, [3]}, [4]} which studies agents (artificial or human) who have to produce messages in order to express well defined meanings. To estimate the compositionality of a set of message-meaning pairs, one can compute a Pearson or Spearman correlation score between textual distances and meaning distances.
Additionally, a permutation test (over random permutation of the message-meaning assignment) can produce a p-value indicating whether the correlation is statistically significant as well as a z-score that quantifies in number of standard deviations the difference between the correlation found in the language and the correlation of random assignments.
This is done among others by Kirby2015CompressionAC, gutierrez-etal-2016-finding, or spike2017minimal-thesis using a Mantel test [5]}, which is based on Pearson's notion of correlation.
